Chapter 6, Problem 68QP

Summary Introduction

To determine: Whether the policy is worth for purchasing it.

Introduction:

The future value of cash flow is the accumulated value including the interest after a specified period. It is utilized to take effective decision at present or to assess the investment potentiality. The total of the future values in each cash flow is the multiple cash flow.

Answer to Problem 68QP

The policy is not worth for purchasing it.

Explanation of Solution

Given information:

An insurance company offers a new policy to their customers. They policy is bought by the children parents or grandparents at the time of the child’s birth. The parents can make 6 payments to the insurance company, and the six payments are as follows:

• On the 1^st birthday the payment amount is $750.
• On the 2^nd birthday the payment amount is $750.
• On the 3^rd birthday the payment amount is $850.
• On the 4^th birthday the payment amount is $850.
• On the 5^th birthday the payment amount is $950.
• On the 6^th birthday the payment amount is $950.

After the 6^th birthday of the child, the payment is not made. At the time when the child will be 65 years, he or she will get $250,000. The interest rate for the first 6 years is 10% and for the rest of the years is 7%.

Note: From the given information, it is essential to compute the future value of the premiums for the comparisons of the promised cash payments at 65 years. Thus, it is necessary to determine the premiums value at year 6 first as the rate of interest varies at that time.

Formula to calculate the future value:

$\text{Future value}=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}$

Note: PV denotes the present value, r denotes rate of discount and t denotes the number of years.

Compute the future value for the five years

$\begin{array}{c}{\text{Future value}}_1=\text{PV}{\left(1+r\right)}^t\\ =\$750{\left(1+0.10\right)}^5\\ =\$750{\left(1.10\right)}^5\\ =\$1,207.88\end{array}$

Hence, the future value for the year 1 is $1,207.88.

$\begin{array}{c}{\text{Future value}}_2=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$750{\left(1+0.10\right)}^4\\ =\$750{\left(1.10\right)}^4\\ =\$1,098.08\end{array}$

Hence, the future value for the year 2 is $1,098.08.

$\begin{array}{c}{\text{Future value}}_3=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$850{\left(1+0.10\right)}^3\\ =\$850{\left(1.10\right)}^3\\ =\$1,131.35\end{array}$

Hence, the future value for the year 3 is $1,131.35.

$\begin{array}{c}{\text{Future value}}_4=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$850{\left(1+0.10\right)}^2\\ =\$850{\left(1.10\right)}^2\\ =\$1,028.5\end{array}$

Hence, the future value for the year 4 is $1,028.5.

$\begin{array}{c}{\text{Future value}}_5=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$950\left(1+0.10\right)\\ =\$950\left(1.10\right)\\ =\$1,045\end{array}$

Hence, the future value for the year 5 is $1,045.

Compute the value at the year 6:

$\begin{array}{c}\text{Value at year 6}=\$1,207.88+\$1,098.08+\$1,131.35+\$1,028.5+\$1,045+\$950\\ =\$6,460.81\end{array}$

Note: The value of year 6 is calculated by adding all the computed future values and the 6^th year value is $950.

Hence, the value at year 6 is $6,460.81.

Compute the future value of the lump sum at the 65^th birthday of the child:

$\begin{array}{c}\text{Future value}=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$6,460.81{\left(1+0.07\right)}^{59}\\ =\$6,460.81{\left(1.07\right)}^{59}\\ =\$349,888.65\end{array}$

Note: The number of years is 59, as only after the 6^th birthday of the child the payments are not paid.

Hence, the future value of the lump sum at the 65^th birthday of the child is $349,888.65.

From the above calculation of the future value, it can be stated that the policy is not worth purchasing, as the deposits value at the future is $349,888.65and the contract will pay off $250,000. The premiums amount to $99,888.65 that are more than the payoff of the policy.

Note: The present value of the two cash flows can be compared

Formula to calculate the present value of the premiums:

$\text{Present value}=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}$

Compute the present value of the premiums:

$\begin{array}{c}\text{Present value }=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\$750}{{\left(1.10\right)}^1}+\frac{\$750}{{\left(1.10\right)}^2}+\frac{\$850}{{\left(1.10\right)}^3}+\frac{\$850}{{\left(1.10\right)}^4}+\frac{\$950}{{\left(1.10\right)}^5}+\frac{\$950}{{\left(1.10\right)}^6}\\ =\left(\begin{array}{l}\$681.8181818+\$619.8347107+\$638.6175808+\\ \$580.5614371+\$589.8752569+\$536.2502336\end{array}\right)\\ =\$3,646.96\end{array}$

Hence, the present value of the premiums is $3,646.96.

The today’s value of $250,000 at the age of 65 is as follows:

$\begin{array}{c}\text{Present value}=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\$250,000}{{\left(1+0.07\right)}^{59}}\\ =\frac{\$250,000}{54.1555391}\\ =\$4,616.33\end{array}$

$\begin{array}{c}\text{Present value}=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\$4,616.33}{{\left(1+0.10\right)}^6}\\ =\frac{\$4,616.33}{1.771561}\\ =\$2,605.80\end{array}$

The cash flow of the premiums is still higher. At the time zero, the difference is $1,041.16 $\left(\$3,646.96-\$2,605.80\right)$. At the time of comparing the streams of two or more cash flows, the cash flows with the maximum value at one time will have the highest value at another time.